Aircraft: B767
Phase of flight: Initial Approach
Anomaly: Conflict Airborne Conflict; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Clearance

OE Captain operating 1st leg as Captain after 3 years of downgrade was PF. Upon arrival in terminal area; runway assignment was changed. As PM; Check airman; after receiving runway change with less than 15 miles to airport; I was setting up radios and reviewing approach plate for PF. I acknowledged a clearance with a vector to join the localizer; descend 5000. Both were verified by PF/PM. I observed PF arm the A/P LOC (only) function. Subsequently ATC issued traffic at 11-12 o'clock 3-4 miles; below us; which was crossing our path to land on parallel runway. Traffic was descending. I called the 'traffic in sight' to the approach controller. As we joined the localizer; a number of events happened simultaneously. The glide slope became 'alive'. The PF called for gear and flaps. The PF commented on the close proximity of the traffic (~2.5 miles; ~1000 ft below us and descending). This generated a brief conversation regarding the TCAS display; as well as continued visual monitoring of the traffic. All occurred as we were approaching (descending to) 5000 ft. Apparently when I called the traffic in sight to ATC; the PF thought we had been cleared for the visual approach; and armed the autopilot Approach (GS) mode. I did not observe this; and the PF did not annunciate his action. At 5000 ft we joined the glide slope. We were both watching the traffic crossing in front of us. Separation was reasonable and certainly safe. I was unaware we had 'captured' the glide slope and left 5000 ft (altitude still in MCP window). Since the GS was armed; there was no altitude alert for leaving the MCP altitude. At approximately 4700 ft the Approach Controller told us to 'stop descent'; which we accomplished by 4600 ft. Vertical separation with the traffic appeared to be 1000-1100 ft; horizontal separation was ~3 miles on the TCAS. The Controller told us that our last assigned altitude was 5000 ft (which would have caused us to be considerably above the GS). I reiterated again to the Controller that we had the traffic in sight. Controller subsequently cleared us to 'maintain visual separation'; cleared for the Visual Approach. We continued our approach to a normal landing. Upon clearance for the Visual; the PF made some brief comments that he believed we had already (previously) been cleared for the Visual approach which is why he continued below 5000 ft; with visual separation with the traffic. I halted the conversation to protect sterile cockpit as it was not related to the current tasks. The traffic situation was unusual in the manner which the traffic crossed in front of us; tracking to a parallel runway. Also unusual was a clearance which restricted us to an altitude which would have caused a significant deviation above the glide slope. Workload was high due to the late runway change; ATC instructions; proximity to the traffic; and the fact the Captain was on his first flight. I do not believe we were cleared the Visual approach at the time we descended below 5000 ft; on the glideslope. The PF believes we had received an approach clearance. I would not have allowed us to Arm/join the glideslope if I had been aware of the PF's actions. We switched to Tower frequency almost immediately after we received our approach clearance so there was not time to clarify/follow up with the Approach controller. Their comment certainly made me think we had not received a clearance below 5000 ft. I do not believe there was a 'loss of separation'; nor was there a TCAS TA or RA; but there was an altitude deviation of 300-400 ft. [Report narrative contained no additional information.] B767 flight crew reported descending on approach before being cleared by ATC. Traffic was crossing in front and below for a parallel runway.